Quartz crystal resonator noise and fluctuation-dissipation theorem considerations

Theoretical and experimental investigations of 1/f noise in quartz crystal resonators

“…1). In a previous paper [3], we described these fluctuations by a 1D viscoelastic model of the longitudinal vibrations along the thickness of the quartz resonator (axis x 2 ). The level of white noise was found to be far below what is measured, meaning that viscoelastic damping of thickness fluctuations is not the dominant noise process.…”

“…Theoretical and experimental works are reported in this paper. We first improve on our previous study using fluctuation dissipation theorem (FDT) by adding a term for internal damping [2] to the classical wave equation already studied [3]. This allows us to recover a 1/f noise spectral density at low frequency.…”

Volume dependence in handel's model of quartz crystal resonator noise

“…However, several points seem to have a lower noise than the intrinsic limit supposedly set by Handel's model if one uses  = 1 cm 3 as proposed in Handel's papers. At this point, one could either question this  = 1 cm 3 value, as we did in another paper [8], or use the above-defined acoustic volume, describing the volume in which elastic energy is confined, instead of the geometric volume between the electrodes. For plano-convex resonators, formula (10) was used instead of the volume between electrodes to draw Fig.…”

“…Second, we recall that A. van der Ziel was able to recover the same formula [20], except for the detailed expression of , using semiclassical arguments. However, we should also note that a careful analysis of this detailed analytical expression of  and the way it is demonstrated by Handel calls for more precise definitions of some of the quantities occurring in this expression, so as to have an explicit way to numerically evaluate them more precisely [8].…”

“…Hence, no entirely satisfactory quantitative predictive model can be given in order to optimize the industrial processes or simply understand the spread of characteristics in a single fabrication batch. Two models have indeed attempted to give quantitative predictions of the level of 1/f noise in quartz resonators, the 1/f quantum noise theory of P. Handel [6] and the degenerated thermal boson theory of M. Planat [7], but the prediction of M. Planat was orders of magnitude above the best measurements, maybe because of the use of the traditional phononic density of states despite the fact that all the harmonics of the fundamental modes had already been taken into account in the calculation of the partition function per mode [8].…”